pith. sign in

arxiv: 2506.01110 · v3 · submitted 2025-06-01 · 🪐 quant-ph

Parity-Time Symmetric Spin-1/2 Richardson-Gaudin Models

M. W. AlMasri This is my paper

Pith reviewed 2026-05-19 10:50 UTC · model grok-4.3

classification 🪐 quant-ph
keywords PT symmetryRichardson-Gaudin modelsintegrable spin chainsnon-Hermitian Hamiltoniansquantum dynamicssimilarity transformationmetric operator
0
0 comments X

The pith

PT-symmetric deformations keep Richardson-Gaudin spin models integrable and deliver exact spin dynamics via a metric operator.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper deforms the standard closed Richardson-Gaudin Hamiltonian for spin-1/2 particles by introducing complex transverse fields and coupling strengths while preserving a PT symmetry defined by parity as the product of all sigma_z operators and a time-reversal that flips only the y-component. The deformed model stays integrable because its conserved charges continue to satisfy generalized commutativity relations. A similarity transformation converts the non-Hermitian Hamiltonian into an equivalent Hermitian one, with the metric operator rho equal to the exponential of minus the sum of q_i times S_i^z serving as the operator that defines the physical inner product. Numerical spectra display the expected PT structure of real eigenvalues or complex-conjugate pairs, and exact closed-form expressions for the time evolution are obtained that show undamped coherent oscillations when PT symmetry remains unbroken and exponentially modulated behavior once it breaks.

Core claim

PT-symmetric Richardson-Gaudin Hamiltonians for spin-1/2 systems remain integrable after deformation by complex parameters. The parity operator is the global product of sigma_z and the time-reversal operator reverses only the y-component of each spin. The metric operator rho = exp(-sum q_i S_i^z) produces a positive-definite inner product, the conserved charges obey generalized commutativity, and the time-dependent spin expectation values admit exact analytic expressions that remain oscillatory in the unbroken PT phase.

What carries the argument

The metric operator rho = e^{-sum q_i S_i^z} that supplies the physical inner product and enables the similarity transformation to a Hermitian Hamiltonian while keeping the conserved charges intact.

If this is right

  • Exact analytic expressions for spin dynamics follow directly from the integrable structure in both unbroken and broken phases.
  • Low-lying eigenstates remain in the unbroken PT phase while higher states may break symmetry.
  • The generalized commutativity of the charges guarantees the existence of a complete set of commuting operators despite non-Hermiticity.
  • Coherent oscillations persist in the unbroken phase and become exponentially modulated once symmetry breaks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same deformation technique could be applied to other families of integrable spin models beyond the Richardson-Gaudin case.
  • The metric construction offers an alternative route to non-Hermitian dynamics that does not rely on Lindblad master equations.
  • Quantum simulators with tunable complex couplings could test the predicted transition from coherent to modulated spin evolution.

Load-bearing premise

A similarity transformation exists that maps the non-Hermitian PT-symmetric Hamiltonian onto a Hermitian operator without destroying the integrability structure or the form of the conserved charges.

What would settle it

A numerical diagonalization or exact solution that produces eigenvalues that are neither real nor complex-conjugate pairs would show that the PT symmetry is not realized as claimed.

Figures

Figures reproduced from arXiv: 2506.01110 by M. W. AlMasri.

Figure 1
Figure 1. Figure 1: FIG. 1: Pictorial representation of the [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Comparison of coupling terms Γ [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The eigenvalues of the conserved charges [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: , panels (c) and (f), illustrate the dynamics of spins in a dissipative environment under weak (c) and strong (f) coupling regimes. These simulations are performed using the Lindblad master equation within the Markovian approximation of the form: dρ dt = −i[Q1, ρ] +X k  LkρL† k − 1 2 {L † kLk, ρ}  , (112) where Q1 is the Hamiltonian, and Lk are the Lindblad operators describing dissipation. The expectati… view at source ↗
read the original abstract

We construct a $\mathcal{PT}$-symmetric Richardson--Gaudin models for spin-$\tfrac{1}{2}$ systems by deforming the closed integrable Hamiltonian through complex-valued transverse magnetic fields and coupling constants. By defining parity as $\mathcal{P} = \prod_i \sigma_i^z$ and adopting a time-reversal operator that flips only the $y$-component of spin, we establish a consistent $\mathcal{PT}$-symmetric framework distinct from open-system approaches based on Lindblad dynamics. The resulting model remains integrable, with conserved charges satisfying generalized commutativity conditions. We explicitly construct the Hermitian counterpart via a similarity transformation and identify the metric operator $\rho = e^{-\sum_i q_i S_i^z}$ that defines the physical inner product. Numerical diagonalization reveals the characteristic $\mathcal{PT}$ spectral structure: eigenvalues are either real or form complex conjugate pairs, with partial symmetry breaking wherein low-energy states remain in the unbroken phase. We further derive exact analytical expressions for spin dynamics, showing coherent oscillations in the unbroken phase and exponentially modulated behavior in the broken phase.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript constructs PT-symmetric spin-1/2 Richardson-Gaudin models by deforming the closed integrable Hamiltonian with complex-valued transverse magnetic fields and coupling constants. Parity is defined as P = product sigma_i^z and time-reversal flips only the y-component. The resulting model is claimed to remain integrable, with conserved charges satisfying generalized commutativity conditions. A similarity transformation is used to construct the Hermitian counterpart, with the metric operator rho = exp(-sum q_i S_i^z) defining the physical inner product. The paper reports the standard PT spectral structure (real eigenvalues or conjugate pairs) with partial symmetry breaking, and derives exact analytical expressions for spin dynamics showing coherent oscillations in the unbroken phase and exponentially modulated behavior in the broken phase.

Significance. If the integrability preservation via the similarity transformation holds without post-hoc adjustments, this would extend the exactly solvable Richardson-Gaudin class into the non-Hermitian PT-symmetric regime, providing a concrete metric and closed-form dynamics that could be useful for modeling coherent phenomena in PT-symmetric spin chains. The explicit metric form and analytical dynamics are potential strengths if the generalized commutativity is rigorously verified.

major comments (2)
  1. The central claim that the PT-symmetric deformation preserves integrability rests on a similarity transformation that maps H to a Hermitian operator while leaving the charge algebra intact under the metric rho. However, the explicit form of the deformed conserved charges and the verification that they satisfy the generalized commutativity condition [Q_i, Q_j]_rho = 0 (or equivalent) after the complex deformation is not provided in sufficient detail. This step is load-bearing for both the integrability assertion and the exact spin-dynamics formulas.
  2. The parameters q_i appearing in the metric rho = exp(-sum q_i S_i^z) are introduced to restore Hermiticity for the chosen complex fields and couplings. The manuscript does not demonstrate that these q_i are derived from an external consistency condition or benchmark rather than selected to enforce the PT requirement, which raises a risk of circularity in the construction.
minor comments (2)
  1. The abstract refers to 'numerical diagonalization' revealing the PT spectral structure; specifying the system sizes, basis truncation, or diagonalization routine used would improve reproducibility.
  2. The distinction between this closed-system PT framework and open-system Lindblad approaches could be clarified with a short comparative paragraph in the introduction.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable comments on our manuscript. We address each of the major comments below and outline the revisions we plan to make to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: The central claim that the PT-symmetric deformation preserves integrability rests on a similarity transformation that maps H to a Hermitian operator while leaving the charge algebra intact under the metric rho. However, the explicit form of the deformed conserved charges and the verification that they satisfy the generalized commutativity condition [Q_i, Q_j]_rho = 0 (or equivalent) after the complex deformation is not provided in sufficient detail. This step is load-bearing for both the integrability assertion and the exact spin-dynamics formulas.

    Authors: We agree that additional detail on this point would improve the manuscript. The integrability follows from the fact that the deformed charges are obtained by conjugating the original charges with the metric operator, preserving the algebra in the rho-inner product. In the revised manuscript, we will provide the explicit expressions for the deformed conserved charges and include a verification of the generalized commutativity [Q_i, Q_j]_rho = 0, either in the main text or as an appendix. This will also clarify the foundation for the exact spin-dynamics formulas. revision: yes

  2. Referee: The parameters q_i appearing in the metric rho = exp(-sum q_i S_i^z) are introduced to restore Hermiticity for the chosen complex fields and couplings. The manuscript does not demonstrate that these q_i are derived from an external consistency condition or benchmark rather than selected to enforce the PT requirement, which raises a risk of circularity in the construction.

    Authors: The q_i are determined by solving the system of equations that ensure the similarity-transformed Hamiltonian is Hermitian. For each complex parameter in the deformation, the corresponding q_i is fixed by canceling the non-Hermitian contributions in the effective operator. This is a direct consequence of the PT-symmetric structure and the form of the Richardson-Gaudin integrals of motion. To eliminate any perception of circularity, we will add in the revision the explicit derivation of the q_i from the Hermiticity condition, showing how they are computed from the complex fields and couplings. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper starts from a known Hermitian Richardson-Gaudin Hamiltonian, deforms it via complex transverse fields and couplings to enforce PT symmetry (with P and T defined explicitly), then constructs the similarity transformation and metric operator ρ = e^{-∑ q_i S_i^z} to obtain the Hermitian counterpart while preserving the integrability structure and generalized commutativity of the charges. The spectral structure, unbroken/broken phases, and exact spin-dynamics expressions follow from this explicit construction and numerical verification rather than from any parameter fitted to reproduce a target prediction or from a self-citation that itself assumes the result. No load-bearing step reduces by definition or construction to the input; the derivation remains self-contained.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The construction rests on the existence of a similarity transformation that simultaneously restores Hermiticity and preserves integrability; the q_i parameters in the metric are introduced without independent derivation shown in the abstract.

free parameters (1)
  • q_i
    Coefficients in the metric operator rho = exp(-sum q_i S_i^z); chosen to make the transformed Hamiltonian Hermitian.
axioms (1)
  • domain assumption The deformed Hamiltonian with complex fields and couplings admits a set of conserved charges that satisfy generalized commutativity.
    Invoked to claim integrability after deformation.

pith-pipeline@v0.9.0 · 5717 in / 1302 out tokens · 20728 ms · 2026-05-19T10:50:49.615955+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

64 extracted references · 64 canonical work pages

  1. [1]

    (10) 6 FIG

    Rational Model: In the rational model, the coupling terms are inversely proportional to the energy differences: Γx ij = 1 ϵi − ϵj , Γz ij = 1 ϵi − ϵj . (10) 6 FIG. 2: Comparison of coupling terms Γ x ij and Γz ij for the rational, trigonometric, and hyperbolic models as functions of the energy difference ϵi − ϵj. This model represents the simplest case, w...

    work page

  2. [2]

    (11) Here, the coupling terms exhibit oscillatory behavior, which is characteristic of systems with periodic boundary conditions or angular coordinates

    Trigonometric Model: The trigonometric model introduces periodic behavior through sine and cotangent functions: Γx ij = 1 sin(ϵi − ϵj) , Γz ij = cot(ϵi − ϵj). (11) Here, the coupling terms exhibit oscillatory behavior, which is characteristic of systems with periodic boundary conditions or angular coordinates

    work page

  3. [3]

    (12) This model is often associated with systems exhibiting exponential localization or long-range interactions

    Hyperbolic Model: The hyperbolic model incorporates exponential growth or decay through hyperbolic sine and cotangent functions: Γx ij = 1 sinh(ϵi − ϵj) , Γz ij = coth(ϵi − ϵj). (12) This model is often associated with systems exhibiting exponential localization or long-range interactions. In Fig.2, we plot the coupling terms Γ x ij and Γz ij for the rati...

    work page

  4. [4]

    Coleman, Introduction to Many-Body Physics , Cambridge University Press (2015)

    P. Coleman, Introduction to Many-Body Physics , Cambridge University Press (2015)

    work page 2015

  5. [5]

    Ring , P

    P. Ring , P. Schuck, The Nuclear Many-Body Problem , Springer-Verlag Berlin Heidelberg (1980)

    work page 1980

  6. [6]

    Amico, R

    L. Amico, R. Fazio, A. Osterloh, and V. Vedral, Entanglement in many-body systems , Rev. Mod. Phys. 80, 517, (2008)

    work page 2008

  7. [7]

    Dukelsky, S

    J. Dukelsky, S. Pittel, and G. Sierra, Colloquium: Exactly solvable Richardson-Gaudin models for many-body quantum systems , Rev. Mod. Phys. 76, 643 (2004)

    work page 2004

  8. [8]

    P. W. Claeys, Richardson-Gaudin models and broken integrability , Ph.D. thesis, Ghent Uni- 26 versity (2018)

    work page 2018

  9. [9]

    R. W. Richardson, A restricted class of exact eigenstates of the pairing-force Hamiltonian , Phys. Lett. 3, 277 (1963)

    work page 1963

  10. [10]

    R. W. Richardson and N. Sherman, Exact eigenstates of the pairing-force Hamiltonian , Nucl. Phys. 52, 221 (1964)

    work page 1964

  11. [11]

    R. W. Richardson, Numerical study of the 8-32-particle eigenstates of the pairing Hamiltonian, Phys. Rev. 141, 949 (1966)

    work page 1966

  12. [12]

    R. W. Richardson, Pairing in the limit of a large number of particles , J. Math. Phys. 18, 1802 (1977)

    work page 1977

  13. [13]

    Gaudin, Diagonalisation d’une classe d’hamiltoniens de spin , J.Phys

    M. Gaudin, Diagonalisation d’une classe d’hamiltoniens de spin , J.Phys. 37 (10), 1087 - 1098 (1976)

    work page 1976

  14. [14]

    Gaudin, The Bethe Wavefunction, Cambridge University Press (2014)

    M. Gaudin, The Bethe Wavefunction, Cambridge University Press (2014). Translated by Jean- S´ ebastien Caux

    work page 2014

  15. [15]

    E. K. Sklyanin, Separation of variables in the Gaudin model , J.Math.Sci 47 , 2473–2488 (1989)

    work page 1989

  16. [16]

    Feigin, E

    B. Feigin, E. Frenkel and N. Reshetikhin, Gaudin model, Bethe Ansatz and critical level , Comm. Math. Phys. 166 (1994) 27–62

    work page 1994

  17. [17]

    M. C. Cambiaggio, A. M. F. Rivas, and M. Saraceno, Integrability of the pairing Hamiltonian, Nucl. Phys. A 624, 157 (1997)

    work page 1997

  18. [18]

    Skrypnyk, Separation of Variables, Quasi-Trigonometric r-Matrices and Generalized Gaudin Models, SIGMA 18 (2022), 074, 18 pages

    T. Skrypnyk, Separation of Variables, Quasi-Trigonometric r-Matrices and Generalized Gaudin Models, SIGMA 18 (2022), 074, 18 pages

    work page 2022

  19. [19]

    elliptic

    T. Skrypnyk, Classical r-matrices, “elliptic” BCS and Gaudin-type hamiltonians and spectral problem, Nuclear Physics B 941, 225–248 (2019)

    work page 2019

  20. [20]

    Dimo, and A

    C. Dimo, and A. Faribault, Quadratic operator relations and Bethe equations for spin-1/2 Richardson-Gaudin models, J. Phys. A: Math. Theor. 51 325202 (2018)

    work page 2018

  21. [21]

    W Claeys, C

    P. W Claeys, C. Dimo, S. De Baerdemacker and A. Faribault, Integrable spin- 1/2 Richard- son–Gaudin XYZ models in an arbitrary magnetic field , J. Phys. A: Math. Theor. 52 08LT01 (2019)

    work page 2019

  22. [22]

    Villazon, A

    T. Villazon, A. Chandran, and P. W. Claeys, Integrability and dark states in an anisotropic central spin model, Phys. Rev. Research 2, 032052(R) (2020)

    work page 2020

  23. [23]

    P. W. Claeys and A. Lamacraft, Dissipative dynamics in open XXZ Richardson-Gaudin mod- 27 els, Phys. Rev. Research 4, 013033 (2022)

    work page 2022

  24. [24]

    De Nadai, N

    E. De Nadai, N. Maestracci, and A. Faribault, Integrability and dark states of the XX spin-1 central spin model in a transverse field , Phys. Rev. B 110, 205427 (2024)

    work page 2024

  25. [25]

    C. M. Bender and S. Boettcher, Real Spectra in Non-Hermitian Hamiltonians Having PT- Symmetry, Phys. Rev. Lett. 80, 5243 (1998)

    work page 1998

  26. [26]

    C. M. Bender, S. Boettcher, and P. N. Meisinger, PT- symmetric quantum mechanics , J. Math. Phys. 40, 2201 (1999)

    work page 1999

  27. [27]

    Dorey, C

    P. Dorey, C. Dunning and R. Tateo, Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics , J. Phys. A 34 , L391 (2001)

    work page 2001

  28. [28]

    C. M. Bender, D. C. Brody, H. F. Jones, Complex Extension of Quantum Mechanics , Phys. Rev. Lett. 89, 270401 (2002); Erratum Phys. Rev. Lett. 92, 119902 (2004)

    work page 2002

  29. [29]

    C. M. Bender and D. W. Hook, PT-symmetric quantum mechanics , Rev. Mod. Phys. 96, 045002 (2024)

    work page 2024

  30. [30]

    Guo et al., Observation of PT-Symmetry Breaking in Complex Optical Potentials , Physical Review Letters , 103, 093902 (2009)

    work page 2009

  31. [31]

    Esaki et al., Edge States and Topological Phases in Non-Hermitian Systems , Physical Review B , 84, 205128 (2011)

    work page 2011

  32. [32]

    Cartarius et al., Model of a PT-Symmetric Bose-Einstein Condensate in a δ-Function Double- Well Potential , Physical Review A , 86, 013612 (2012)

    work page 2012

  33. [33]

    G von Gehlen, Critical and off-critical conformal analysis of the Ising quantum chain in an imaginary field, J. Phys. A: Math. Gen. 24 5371 (1991)

    work page 1991

  34. [34]

    Korff, PT symmetry of the non-Hermitian XX spin-chain: non-local bulk interaction from complex boundary fields, J

    C. Korff, PT symmetry of the non-Hermitian XX spin-chain: non-local bulk interaction from complex boundary fields, J. Phys. A: Math. Theor. 41 295206 (2008)

    work page 2008

  35. [35]

    O. A. Castro-Alvaredo, and A. Fring, A spin chain model with non-Hermitian interaction: The Ising quantum spin chain in an imaginary field , J. Phys. A 42 :465211 (2009)

    work page 2009

  36. [36]

    Galda and V

    A. Galda and V. M. Vinokur, Parity-time symmetry breaking in spin chains , Phys. Rev. B 97, 201411 (2018)

    work page 2018

  37. [37]

    Prosen, PT-Symmetric Quantum Liouvillean Dynamics , Phys

    T. Prosen, PT-Symmetric Quantum Liouvillean Dynamics , Phys. Rev. Lett. 109, 090404 (2012)

    work page 2012

  38. [38]

    G. A. Starkov, M. V. Fistul, I. M. Eremin, Quantum phase transitions in non-Hermitian PT- symmetric transverse-field Ising spin chains , Annals of Physics Volume 456, 169268 (2023). 28

    work page 2023

  39. [39]

    M. W. AlMasri, M. R.B. Wahiddin, Integral Transforms and PT-symmetric Hamiltonians , Chinese Journal of Physics, 85, Pages 127-134 (2023)

    work page 2023

  40. [40]

    Fring, PT-symmetric deformations of integrable models , Phil

    A. Fring, PT-symmetric deformations of integrable models , Phil. Trans. R. Soc. A 371 20120046 (2013)

    work page 2013

  41. [41]

    Gardas, S

    B. Gardas, S. Deffner, and A. Saxena, PT -symmetric slowing down of decoherence , Phys. Rev. A 94, 040101 (R) (2016)

    work page 2016

  42. [42]

    Cen and A

    J. Cen and A. Saxena, Anti- PT symmetric qubit: Decoherence and entanglement entropy , Phys. Rev. A 105, 022404 (2022)

    work page 2022

  43. [43]

    T. E. Lee, Y. N. Joglekar, PT-symmetric Rabi model: Perturbation theory , Phys. Rev. A 92, 042103 (2015)

    work page 2015

  44. [44]

    Lu, Hui Li, J-K

    X. Lu, Hui Li, J-K. Shi, L-B. Fan, V. Mangazeev, Z-M. Li, M. T. Batchelor, PT-symmetric quantum Rabi model, Phys. Rev. A 108, 053712 (2023)

    work page 2023

  45. [45]

    M. W. AlMasri, Supersymmetry of PT-symmetric tridiagonal Hamiltonians , Mod. Phys. Lett. A 36 (35) , (2021)

    work page 2021

  46. [46]

    Ghatak, T

    A. Ghatak, T. Das, Theory of superconductivity with non-Hermitian and parity-time reversal symmetric cooper pairing symmetry, Phys. Rev. B 97, 014512 (2018)

    work page 2018

  47. [47]

    Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems , J

    I. Rotter, A non-Hermitian Hamilton operator and the physics of open quantum systems , J. Phys. A: Math. Theor. 42 153001 (2009)

    work page 2009

  48. [48]

    E. B. Davies, Quantum Theory of Open Systems , Academic Press, 1976

    work page 1976

  49. [49]

    Breuer and F

    H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems , Oxford University Press, 2002

    work page 2002

  50. [50]

    Rivas and S

    A. Rivas and S. F. Huelga, Open Quantum Systems: An Introduction , SpringerBriefs in Physics, Springer, 2012

    work page 2012

  51. [51]

    Fazio, J

    R. Fazio, J. Keeling, L. Mazza, M. Schir` o,Many-Body Open Quantum Systems, Lecture Notes arXiv:2409.10300v4 (2024)

    work page arXiv 2024

  52. [52]

    Babelon, and D

    O. Babelon, and D. Talalaev, On the Bethe Ansatz for the Jaynes-Cummings-Gaudin model , J. Stat. Mech. P06013 (2007)

    work page 2007

  53. [53]

    L. D. Faddeev and L. A. Takhtajan, Hamiltonian Methods in the Theory of Solitons , Springer Series in Soviet Mathematics, Springer-Verlag, 1987

    work page 1987

  54. [54]

    R. J. Baxter, Exactly Solved Models in Statistical Mechanics , Academic Press, 1982

    work page 1982

  55. [55]

    V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and 29 Correlation Functions, Cambridge University Press, 1993

    work page 1993

  56. [56]

    Ortiz, R

    G. Ortiz, R. Somma, J. Dukelsky, and S. Rombouts, Exactly-solvable models derived from a generalized Gaudin algebra, Nucl. Phys. B 707, 421 (2005)

    work page 2005

  57. [57]

    Shankar, Principles of Quantum Mechanics , 2nd edition, Plenum Press (1994)

    R. Shankar, Principles of Quantum Mechanics , 2nd edition, Plenum Press (1994)

    work page 1994

  58. [58]

    P. D. Mannheim, Extension of the CPT theorem to non-Hermitian Hamiltonians and unstable states, Physics Letters B 753, Pages 288-292, (2016)

    work page 2016

  59. [59]

    Beetar, N

    C. Beetar, N. Gupta, S. Shajidul Haque, J. Murugan, H. J. R. Van Zyl, Complexity and Operator Growth for Quantum Systems in Dynamic Equilibrium , J. High Energ. Phys. 2024, 156 (2024)

    work page 2024

  60. [60]

    C. Liu, H. Tang, and H. Zhai, Krylov complexity in open quantum systems , Phys. Rev. Re- search 5 , 033085 (2023)

    work page 2023

  61. [61]

    Y. Zhou, W. Xia, L. Li, W. Lim, Diagnosing Quantum Many-body Chaos in Non-Hermitian Quantum Spin Chain via Krylov Complexity , arXiv:2501.15982 (2025)

    work page arXiv 2025

  62. [62]

    Medina-Guerra, I

    E. Medina-Guerra, I. V. Gornyi, Y. Gefen, Correlations and Krylov spread for a non-Hermitian Hamiltonian: Ising chain with a complex-valued transverse magnetic field , arXiv:2502.07775 (2025)

    work page arXiv 2025

  63. [63]

    Buluta and F

    I. Buluta and F. Nori, Quantum Simulators , Science Vol 326, Issue 5949 , pp. 108-111 (2009)

    work page 2009

  64. [64]

    I. M. Georgescu, S. Ashhab, and F. Nori, Quantum simulation , Rev. Mod. Phys. 86 , 153 (2014). 30

    work page 2014